Applied mathematics often operates by way of shakily rationalizedexpedients that can neither be understood in a deductive-nomological nor in an anti-realist setting.Rather do these complexities, so a recent paper of Mark Wilson argues, indicate some element in ourmathematical descriptions that is alien to the physical world. In this vein the mathematical opportunistopenly seeks or engineers appropriate conditions for mathematics to get hold on a given problem.Honest mathematical optimists, instead, try to liberalize mathematical ontology so as to include all physicalsolutions. Following (...) John von Neumann, the present paper argues that the axiomatization of a scientifictheory can be performed in a rather opportunistic fashion, such that optimism and opportunism appear as twomodes of a single strategy whose relative weight is determined by the status of the field to beinvestigated. Wilson's promising approach may thus be reformulated so as to avoid precarious talk about a physicalworld that is void of mathematical structure. This also makes the appraisal of the axiomatic method inapplied matthematics less dependent upon foundationalist issues. (shrink)

Because of the complex interdependence of physics and mathematics their relation is not free of tensions. The paper looks at how the tension has been perceived and articulated by some physicists, mathematicians and mathematical physicists. Some sources of the tension are identified and it is claimed that the tension is both natural and fruitful for both physics and mathematics. An attempt is made to explain why mathematical precision is typically not welcome in physics.

This paper has two parts, a historical and a systematic. In the historical part it is argued that the two major axiomatic approaches to relativistic quantum field theory, the Wightman and Haag-Kastler axiomatizations, are realizations of the program of axiomatization of physical theories announced by Hilbert in his 6th of the 23 problems discussed in his famous 1900 Paris lecture on open problems in mathematics, if axiomatizing physical theories is interpreted in a soft and opportunistic sense suggested in 1927 by (...) Hilbert, Nordheim, and von Neumann. To show this, Section 2 recalls first some of Hilbert's views about the axiomatic approach to physical theories as these were formulated in his 6th problem. It will be .. (shrink)

By quoting extensively from unpublished letters written by John von Neumann to Garret Birkhoff during the preparatory phase (in 1935) of their ground-breaking 1936 paper that established quantum logic, the main steps in the thought process leading to the 1936 Birkhoff–von Neumann paper are reconstructed. The reconstruction makes it clear why Birkhoff and von Neumann rejected the notion of quantum logic as the projection lattice of an infinite dimensional complex Hilbert space and why they postulated in their 1936 paper that (...) the quantum propositional system should be isomorphic to an abstract projective geometry. Looking at the paper now I see, that I forgot to say this, which should be said somewhere in the first §: That while common logics did apply to quantum mechanics, if the notion of simultaneous measurability is introduced as an auxiliary notion, we wished to construct a logical system, which applies directly to quantum mechanics – without any extraneous secondary notions like simultaneous measurability. And in order to have such a consequent, one-piece system of logics, we must change the classical class calculus of logics. (J. von Neumann to G. Birkhoff, November 21, 1935). (shrink)

InThere Are No Such Things As Theories(French 2020), the reification of theories is critically analysed and rejected. My aim here is to tease out some of the implications of this approach first of all, for how we, philosophers of science, should view the history of science; secondly, for how we should understand the devices that we use in our own philosophical practices; and thirdly, for how we might think about the relationship between the history of science and the philosophy of (...) science. (shrink)

A synaptic algebra is a generalization of the self-adjoint part of a von Neumann algebra. In this article we extend to synaptic algebras the type-I/II/III decomposition of von Neumann algebras, AW∗-algebras, and JW-algebras.

A symmetric monoidal category naturally arises as the mathematical structure that organizes physical systems, processes, and composition thereof, both sequentially and in parallel. This structure admits a purely graphical calculus. This paper is concerned with the encoding of a fixed causal structure within a symmetric monoidal category: causal dependencies will correspond to topological connectedness in the graphical language. We show that correlations, either classical or quantum, force terminality of the tensor unit. We also show that well-definedness of the concept of (...) a global state forces the monoidal product to be only partially defined, which in turn results in a relativistic covariance theorem. Except for these assumptions, at no stage do we assume anything more than purely compositional symmetric-monoidal categorical structure. We cast these two structural results in terms of a mathematical entity, which we call a causal category . We provide methods of constructing causal categories, and we study the consequences of these methods for the general framework of categorical quantum mechanics. (shrink)

Scientific change has two important dimensions: conceptual change and structural change. In this paper, I argue that the existence of conceptual change brings serious difficulties for scientific realism, and the existence of structural change makes structural realism look quite implausible. I then sketch an alternative account of scientific change, in terms of partial structures, that accommodates both conceptual and structural changes. The proposal, however, is not realist, and supports a structuralist version of van Fraassen’s constructive empiricism (structural empiricism).

Mathematical concepts play at least three roles in the application of mathematics: an inferential role, a representational role, and an expressive role. In this paper, I argue that, despite what has often been alleged, platonists do not fully accommodate these features of the application of mathematics. At best, platonism provides partial ways of handling the issues. I then sketch an alternative, anti-realist account of the application of mathematics, and argue that this account manages to accommodate these features of the application (...) process. In this way, a better account of mathematical applications is, in principle, available. (shrink)

A novel conceptual framework for theoretical psychology is presented and illustrated for the example of bistable perception. A basic formal feature of this framework is the non-commutativity of operations acting on mental states. A corresponding model for the bistable perception of ambiguous stimuli, the Necker–Zeno model, is sketched and some empirical evidence for it so far is described. It is discussed how a temporal non-locality of mental states, predicted by the model, can be understood and tested.

The paper sketches an argument against modal monism, more specifically against the reduction of physical possibility to metaphysical possibility. The argument is based on the non-categoricity of quantum logic.

In this paper, I argue that symmetry principles in physics (in particular, in quantum mechanics) have a methodological character, rather than an ontological or an epistemological one. First, I provide a framework to address three related issues regarding the notion of symmetry: (i) how the notion can be characterized; (ii) one way of discussing the nature of symmetry principles, and (iii) a tentative account of some types of symmetry in physics. To illustrate how the framework functions, I then consider the (...) case of the early formulation of quantum mechanics, examining the different roles played by symmetry in this context. Finally, I raise difficulties for ontological and purely episte. (shrink)

This work is a conceptual analysis of certain recent developments in the mathematical foundations of Classical and Quantum Mechanics which have allowed to formulate both theories in a common language. From the algebraic point of view, the set of observables of a physical system, be it classical or quantum, is described by a Jordan-Lie algebra. From the geometric point of view, the space of states of any system is described by a uniform Poisson space with transition probability. Both these structures (...) are here perceived as formal translations of the fundamental twofold role of properties in Mechanics: they are at the same time quantities and transformations. The question becomes then to understand the precise articulation between these two roles. The analysis will show that Quantum Mechanics can be thought as distinguishing itself from Classical Mechanics by a compatibility condition between properties-as-quantities and properties-as-transformations. -/- Moreover, this dissertation shows the existence of a tension between a certain "abstract way" of conceiving mathematical structures, used in the practice of mathematical physics, and the necessary capacity to specify particular states or observables. It then becomes important to understand how, within the formalism, one can construct a labelling scheme. The “Chase for Individuation” is the analysis of different mathematical techniques which attempt to overcome this tension. In particular, we discuss how group theory furnishes a partial solution. (shrink)

72 1024x768 This paper presents evidence from the fields of cognitive science and quantum information theory suggesting quantum theory to be the dominant fundamental logic in the natural world, in direct challenge to the long-held assumption that quantum logic only need be considered ‘in the quantum realm.' A summary of the evolution of quantum logic and quantum theory is presented, along with an overview for the necessity of incomplete quantum knowledge, and some representative aspects of quantum logic. A case can (...) be made that classical logic and theory is a subset of quantum logic and theory, given that elements of quantum physics exist that can never admit classical understanding, including: Bell's theorem, Hardy's theorem, and the Pusey-Barrett-Rudolph theorem. Support can be found for the primacy of quantum logic in the natural world in the cognitive sciences, where recent research studies recognize quantum logic in studies of: the subconscious, decisions involving unknown interconnected variables, memory, and question sequencing. Normal 0 false false false EN-US X-NONE X-NONE. (shrink)